Extremal bounds for Gaussian trace estimation

TL;DR

This paper establishes extremal tail bounds for Gaussian trace estimators on symmetric matrices, relating spectral properties to estimator performance, with implications for matrices with bounded effective rank or fixed norms.

Contribution

It introduces a spectral partial ordering to determine worst-case tail bounds for Gaussian trace estimators on specific matrix families.

Findings

Tail bounds depend on eigenvalue ordering

Extremal bounds characterized for matrices with bounded effective rank

Results applicable to matrices with fixed Frobenius norm and bounded 2-norm

Abstract

This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.